Problem 18 - Algebra and Trigonometry

Question

Find each product and write the result in standard form. $$ (-7-i)(-7+i) $$

Step-by-Step Solution

Answer

The product of (-7-i)(-7+i) is 50.

Step 1 - Apply Formula

The first step is to apply the formula for the product of two complex numbers: \( (a+bi)(c+di) = ac - bd + (ad+bc)i \). In this case, \( a=c=-7 \) and \( b=d=-1 \). Therefore, the formula becomes: \( (-7 -1i)(-7+1i)=-7*(-7) - (-1)*1 + (-7*1-(-1*-7))i \)

Step 2 - Compute Real Part

The next step is to compute the real part of the result. Using the first half of the product: \( -7*(-7)-(-1)*1 \), we get: \( 49-(-1)=49 + 1 = 50 \).

Step 3 - Compute Imaginary Part

Then, compute the imaginary part of the result. Using the second half of the product: \( -7*1 - (-1*-7) \), we get: \( -7 -7 = -14 \). Since this is added to \( -14i \) in the formula, it cancels out, resulting in zero.

Step 4 - Write Result in Standard Form

Hence the result of the product is already in standard form, which is simply the real part computed in step 2, as there is no imaginary term.